Question: Determine how many solutions exist for the system of equations. ${-4x+2y = 6}$ ${-5x+y = -2}$
Answer: Convert both equations to slope-intercept form: ${-4x+2y = 6}$ $-4x{+4x} + 2y = 6{+4x}$ $2y = 6+4x$ $y = 3+2x$ ${y = 2x+3}$ ${-5x+y = -2}$ $-5x{+5x} + y = -2{+5x}$ $y = -2+5x$ ${y = 5x-2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+3}$ ${y = 5x-2}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.